Just use Signal, the privacy first messaging system.
What’s in it for Signal? They’re a non-profit. Money isn’t their main goal. Read about their very interesting history on Wikipedia.
(But if they ever break that trust I’m going back to SMS texts!)
Update a few days later: This post was obviously prompted by the move WhatsApp made to change their privacy policy and ultimatum they gave users to “accept it or go away”. They have since announced the changes won’t take effect until May this year.
This Gizmodo post is the best yet at explaining what changes are to come. It’s a long article worth reading but in summary (and I’m open to correction if I get this wrong), your messages to friends and family were never in danger of being revealed to Facebook. Businesses who use WhatsApp have a special way of using the platform (a business API) and they may use an outside agency to help them deal with this messaging. As soon as that happens the outside agency can see the messages back forth between you and the business so the messages could in theory be stored by that agency. Someone else can see those messages then.
What I’m left wondering about is if companies have been doing this for years already why is a change in the privacy policy required now? Is it because a business can manage this messaging through Facebook.com?
I checked and I didn’t opt out of WhatsApp sending “anonymous data” (which is never really anonymous) to Facebook in 2016 for whatever reason. Maybe I didn’t start using WhatsApp until after this date. I wonder if the GDPR stops this data transfer?